As one learns more about feedback and control systems, at some point, comes an important realization that a variety of distinct problems can be answered if we know how to answer the question: “is some closed-loop system asymptotically stable?” Indeed, the study of stability dominates Chapters 6 and 7, arguably the densest chapters in the book. Many students, when asked what they remember from their undergraduate control class, will quickly point at the *root-locus* or at the *Nyquist stability criterion*. Yet, in the beginning, it might seem that stability is a side dish and not the main attraction.

Take for example Chapter 4. It is easy to overlook that asymptotic stability is a requirement for tracking. Or to mistake internal stability as something that is only important in the *rare* case of pole-zero cancellations. It is common for students to wonder what to do with the system inputs when asked to analyze a feedback system using the root-locus method or the Nyquist stability criterion. For example, here is a common question:

How do you incorporate a constant disturbance into your $L$ transfer function? In the case of this block diagram(apologize for the messiness), I was unable to separate an alpha and C(s) from the K(s) since there is a constant omega [$w$] which would be added to the output of e*K(s). So in this case, do we just ignore omega [$w$]?

Anonymous

The above question reveals some apprehension about “ignoring” the inputs, as if one would be answering *only * the stability question. It also reveals that the student is likely missing a very important point: the reason for reformulating the problem as a stability question is not to *ignore* the inputs but rather to look for answers that hold for *all* possible “well behaved” inputs. Stability provides broad guarantees rather than specific answers.

It might be especially difficult to “let the inputs go” if they have physical meaning. Take for example a problem such as P6.34-P6-36, in which students are asked to revisit the design of a controller for the temperature, $T$, of a water heater using the root-locus method. A model for the system is the differential equation

$$\dot{T}(t)+\left (\frac{v}{m} + \frac{1}{m c R} \right ) T(t) = \frac{1}{m c} \left (u(t) + w(t)\right ) $$

in which the “disturbance” is the very physical signal

$$w(t) = \bar{w}, \quad \bar{w} = v \, c \, T_i +\frac{1}{R} T_0,$$

which is related to things like the system’s thermal properties, $c$ and $R$, and the (constant) in/out flow and its temperature, $v$, $T_0$, $T_i$, which clearly make $\bar{w} \neq 0$. In the presence of a model like this, it is not uncommon to hear someone say: “It doesn’t make any sense to analyse this system if the input $w(t)=\bar{w}$ is not present!”

Yet, because stability allows you to draw conclusions about a large class of inputs, namely all bounded inputs, which $w(t) = \bar{w}$ is certainly a member of, it is safe to go ahead and ask whether a controller will stabilize the closed-loop system without worrying about the particular input $w(t) =\bar{w}$.

Likewise, a question such as “does the closed-loop controller track a given constant reference $\bar{y}$?” can be answered by adding a pole at the origin to the controller (see Lemma 4.1) and assessing asymptotic stability of the resulting closed-loop transfer-functions.

Of all the *tricks* you learn with feedback control this is one of my favorites. So next time, just let your inputs go! Stability will take care of them.

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